On the definition of spin current and the generalization of Kubo formula for linear response to non-abelian fields You-Quan Li Zhejiang Institute of Modern Physics, Zhejiang University, Hangzhou, 310027, P.R. China   Spin and charge currents in systems with Rashba or Dresselhaus spin-orbit couplings are formulated in a unified version of four-dimensional SU(2)$\times$U(1) gauge theory. Similar to the Maxwell field inducing forces on charge and charge current, we derive forces acting on spin and spin current induced by the Yang Mills fields such as the Rashba and Dresselhaus fields and the sheer strain field. The traditional Kubo formula is generalized to describe the linear response with respect to non-Abelian fields. To fulfil the demand for studying spin transport, the SU(2) Kubo formulae are derived by two conventional approaches with different gauge fixings. Those two approaches are shown to be equivalent where the non-conservation of the SU(2) current plays an essential role in guaranteeing the consistency. Some concrete examples relating Spin Hall Effect are considered. The dc spin conductivity vanishes in the system with parabolic unperturbed dispersion relation. By applying a time-dependent Rashba field, the spin conductivity can be measured directly. Our formula is also applied to the high-dimensional representation for the interests of some important models, such as Luttinger model and bilayer Spin Hall system.